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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

An extended Vlasov–Fokker–Planck approach for kinetic simulations of laser plasmas

Vlasov–Fokker–Planck simulation codes occupy an important niche in modeling laser-produced plasmas, since they are well suited to studying the effect of collisions on electron kinetic phenomena, especially energy transport. One of the most important elements of energy transport is the absorption of laser light by the plasma; however, simulating this in detail requires resolving oscillations of the laser light, whose characteristic timescale is orders of magnitude shorter than the simulation time needed to study transport physics. For this reason, most Vlasov–Fokker–Planck codes used to study electron transport in laser plasmas rely on simplified models of the laser–plasma coupling. Their underlying assumptions nominally preclude their use for modeling laser light having short-scale structure in space or time, such as broadband lasers. In this work, we derive a more general computational framework suitable for arbitrarily structured laser fields. Furthermore, our approach is based on an extended set of Vlasov–Fokker–Planck equations that separately solve for the low- and high-frequency plasma response. We implement these extended Vlasov–Fokker–Planck equations in the spherical harmonic code K2 and demonstrate the performance of the method on several laser absorption test problems, with particular attention to the judicious selection of time steps, time integrators, and spherical harmonic truncation, according to the intensity and spectrum of the laser light under consideration. Comparison with the widely used Langdon absorption operator shows the Langdon operator performs remarkably well for predicting laser heating in the simple cases considered here, even in situations that would seem to violate its underlying assumptions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Graph-learning approach to combine multiresolution seismic velocity models

SUMMARY The resolution of velocity models obtained by tomography varies due to multiple factors and variables, such as the inversion approach, ray coverage, data quality, etc. Combining velocity models with different resolutions can enable more accurate ground motion simulations. Toward this goal, we present a novel methodology to fuse multiresolution seismic velocity maps with probabilistic graphical models (PGMs). The PGMs provide segmentation results, corresponding to various velocity intervals, in seismic velocity models with different resolutions. Further, by considering physical information (such as ray path density), we introduce physics-informed probabilistic graphical models (PIPGMs). These models provide data-driven relations between subdomains with low (LR) and high (HR) resolutions. Transferring (segmented) distribution information from the HR regions enhances the details in the LR regions by solving a maximum likelihood problem with prior knowledge from HR models. When updating areas bordering HR and LR regions, a patch-scanning policy is adopted to consider local patterns and avoid sharp boundaries. To evaluate the efficacy of the proposed PGM fusion method, we tested the fusion approach on both a synthetic checkerboard model and a fault zone structure imaged from the 2019 Ridgecrest, CA, earthquake sequence. The Ridgecrest fault zone image consists of a shallow (top 1 km) high-resolution shear-wave velocity model obtained from ambient noise tomography, which is embedded into the coarser Statewide California Earthquake Center Community Velocity Model version S4.26-M01. The model efficacy is underscored by the deviation between observed and calculated traveltimes along the boundaries between HR and LR regions, 38 per cent less than obtained by conventional Gaussian interpolation. The proposed PGM fusion method can merge any gridded multiresolution velocity model, a valuable tool for computational seismology and ground motion estimation.

Geochemistry & Geophysics↗
DOE OSTI2022

Calibrating uncertain parameters in melt pool simulations of additive manufacturing

Melt pool scale numerical modeling of additive manufacturing (AM) processes can provide predictive capabilities and theoretical insight into the process-property-structure-performance relationships for AM parts. Despite capabilities of numerical models to solve complex multi-physics problems, it is often important to consider a tradeoff between detailed physics and computational cost. Therefore, sources of uncertainty in both experimental conditions and the parameters needed for modeling require models to be validated against empirical evidence. Here, a method is proposed to calibrate uncertain parameters used in continuum-scale melt pool models for powder bed fusion (PBF) AM. Both a simplified heat transfer model and a heat transfer and fluid flow model were investigated. A surrogate model and Markov chain-based optimization algorithm calibrated melt pool geometry for models within experimental variation of the target melt pool width and depth from the NIST AM-Bench 2018-02 dataset. The melt pool temperature distributions, solidification parameters, and simulated multi-layer solidification microstructures were compared between the two models. Similar results from both models indicate that calibrated, lower fidelity numerical models may be used in place of higher fidelity models to generate melt pool solidification data. Finally, these calibrated models therefore enable lower computational cost melt pool simulations without a noticeable decrease in simulation accuracy for grain-scale microstructure simulations.

36 MATERIALS SCIENCE↗
DOE OSTI2021

MULTICOM2 open-source protein structure prediction system powered by deep learning and distance prediction

Protein structure prediction is an important problem in bioinformatics and has been studied for decades. However, there are still few open-source comprehensive protein structure prediction packages publicly available in the field. In this paper, we present our latest open-source protein tertiary structure prediction system—MULTICOM2, an integration of template-based modeling (TBM) and template-free modeling (FM) methods. The template-based modeling uses sequence alignment tools with deep multiple sequence alignments to search for structural templates, which are much faster and more accurate than MULTICOM1. The template-free (ab initio or de novo) modeling uses the inter-residue distances predicted by DeepDist to reconstruct tertiary structure models without using any known structure as template. In the blind CASP14 experiment, the average TM-score of the models predicted by our server predictor based on the MULTICOM2 system is 0.720 for 58 TBM (regular) domains and 0.514 for 38 FM and FM/TBM (hard) domains, indicating that MULTICOM2 is capable of predicting good tertiary structures across the board. It can predict the correct fold for 76 CASP14 domains (95% regular domains and 55% hard domains) if only one prediction is made for a domain. The success rate is increased to 3% for both regular and hard domains if five predictions are made per domain. Moreover, the prediction accuracy of the pure template-free structure modeling method on both TBM and FM targets is very close to the combination of template-based and template-free modeling methods. This demonstrates that the distance-based template-free modeling method powered by deep learning can largely replace the traditional template-based modeling method even on TBM targets that TBM methods used to dominate and therefore provides a uniform structure modeling approach to any protein. Finally, on the 38 CASP14 FM and FM/TBM hard domains, MULTICOM2 server predictors (MULTICOM-HYBRID, MULTICOM-DEEP, MULTICOM-DIST) were ranked among the top 20 automated server predictors in the CASP14 experiment. After combining multiple predictors from the same research group as one entry, MULTICOM-HYBRID was ranked no. 5. The source code of MULTICOM2 is freely available at https://github.com/multicom-toolbox/multicom/tree/multicom_v2.0 .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Enhancing Oxygen Stability in Low-Cobalt Layered Oxide Cathode Materials by Three-Dimensional Targeted Doping

In this project, we propose to develop a new concept and a generic platform that can lead to the greatly enhanced stabilization of all high-energy cathode materials, and in particular high-nickel (Ni) and low-cobalt (Co) oxides. The new concept is a 3D doping technology that hierarchically combines surface and bulk doping. We will use surface doping to stabilize the surface of primary particles and also introduce dopants in the bulk to further enhance oxygen stability, conductivity, and structural stability in low-Co oxides under high voltage and deep discharging operating conditions. This new concept not only will deliver a low-cost, high-energy cathode but also will provide a generic method that can stabilize all high-energy cathodes. The proposed novel 3D doping approach is poised to resolve some longstanding challenges in fundamental doping effects on battery materials as well as to reduce Li-ion batteries’ cost and improve their safety, energy density, and lifetime. To tackle this problem, we have formed a highly complementary multi-university/national labs/industry team to enable a doping-central and systematic investigation of low-Co materials and create a knowledge base for many electrode materials to be used in advanced electric vehicles. The successful execution of the proposed project relies on five components that can be carried out by the complementary team members: (1) a theoretical investigation of the surface and bulk stabilizing dopants (Persson), (2) a precise synthesis of materials with targeted doping (Lin and Xin), (3) development of electrolytes for high-Ni low-Co oxides (Xu), (4) multi-scale characterization of the structures and their interfaces by scanning transmission electron microscopy (SEM) and synchrotron X-ray imaging and spectroscopy tools (Xin and Lin), and (5) pouch cell-level integration (Fan). The UCI-led project will enable a doping-central and systematic investigation of low-Co materials and create a knowledge base for many electrode materials to be used in advanced electric vehicles.

25 ENERGY STORAGE↗
DOE OSTI2024

Composite coatings from polycarbosilane derived SiC and Al/SiC cermet active fillers as protective barriers against steel corrosion

Stainless steel is used throughout the world as a structural material. However, it undergoes corrosion damage when exposed to extremely corrosive media, such as the marine environment. An alternative to solve this problem lies in the development of coatings that can withstand extreme conditions but also be easily deposited with inherently corrosion-resistant materials such as silicon carbide (SiC). The present study shows a simple method to produce Al/SiC cermet powders by attrition milling. The resulting cermet powders with a metallic matrix and hemispherical morphology, were employed as fillers in polycarbosilane (PCS) solutions that were sprayed on A304 stainless steel substrates. Al/SiC composite coatings were produced after heating the sprayed suspensions at 700 °C for 1 h in Ar atmosphere. The resulting composite coatings exhibited low surface energies (< 35 mN/m), water contact angles of 53°, and adhesion strength of up to 30 MPa. Finally, corrosion tests were performed in a cyclic corrosion test chamber, showing that these coatings effectively reduced the corrosion rate of stainless steel by 87%, reaching corrosion rate values of 0.007 g/cm 2 year.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Emerging Potassium Metal Anodes: Perspectives on Control of the Electrochemical Interfaces

Potassium metal acts as the anode in emerging potassium metal batteries (KMBs). It further serves as the counter-electrode for potassium ion battery (KIB) half-cells, with its reliable performance being critical for assessing the working electrode material. This first-of-its-kind critical review focuses on the dual challenge of controlling the potassium metal-substrate and the potassium metal-electrolyte interface so as to prevent dendrites. The discussion begins with a comparison of the physical and chemical properties of K metal anodes versus the much oft studied Li and Na metal anodes. Based on established descriptions for root causes of dendrites, the problem should be less severe for K than for Li or Na, while in fact the opposite is observed. The key reason that the K metal surface rapidly becomes dendritic in common electrolytes is its unstable solid electrolyte interphase (SEI). An unstable SEI layer is defined as being non self-passivating. No SEI is perfectly stable during cycling, and all SEI structures are heterogenous both vertically and horizontally relative to the electrolyte interface. The difference between a "stable" and an "unstable" SEI may be viewed as the relative degree to which during cycling it thickens and becomes further heterogeneous. The unstable SEI on K metal leads to a number of interrelated problems, such as low cycling Coulombic efficiency (CE), a severe impedance rise, large overpotentials, and possibly electrical shorting; all of which have been reported to occur as early as in the first ten plating/stripping cycles. Many of the traditional "interface fixes" employed for Li and Na metal anodes, such as various artificial SEIs, surface membranes, barrier layers, the secondary separators, etc. have not been attempted or optimized for the case of K. This is an important area for further exploration, with an understanding that success may come harder than with Li due to K-based SEI reactivity with both ether and ester solvents. The second critical problem with K metal anodes is that they do not thermally or electrochemically wet a standard (untreated) Cu foil current collector. Published experimental and modeling research directly highlights the weak bonding between the K atoms and a Cu surface. Existing surface treatment approaches that achieve improved K wetting are discussed, along with the general design rules for future studies. Also discussed are geometry-based methods to tune nucleation, as well dual approaches where nucleation and SEI structure are tuned through complementary schemes to achieve extended half-cell and full battery stability. We hypothesize that K metal never achieves a planar wetting morphology even at cycle one making the dendrites "baked-in". We propose that classical thin films growth models, Frank van der Merwe (F-M), Volmer Weber (V-W), and Stranski-Krastanov (S-K) can be employed to describe early stage plating behavior. It is demonstrated that island-like V-W growth is the applicable description for the natural plating behavior of K on pristine Cu. Moving forward, there are three inter-related thrusts to be pursued: First, K salt - based electrolyte formulations have to mature and become further tailored to handle the increased reactivity of a metal rather than an ion anode. Second, the K-based SEI structure needs to be further understood and ultimately tuned to be less reactive. Third, the energetics of the K metal - current collector interface must be controlled to promote planar wetting/dewetting throughout cycling.

25 ENERGY STORAGE↗
DOE OSTI2023

Scaling Optimal Allocation of Cloud Resources Using Lagrange Relaxation

The rapid growth of Cloud Computing (CC) has increased the variety of computing resources, storage, and communication services that pose significant new challenges for the efficient use of cloud resources. The cost-efficient allocation of cloud resources has become a decisive premise for the adoption of CC services. The cost-efficient selection and scheduling of these resources to meet the demands of a scientific workflow is a challenging problem that is exacerbated by the inclusion of multiple CC providers. In this paper, we present a novel strategy for the cost-efficient selection of CC resources using Lagrange relaxation. Our approach is based on preselection of resources and demand decomposition to create a series of smaller sub-problems, which allow the estimation of the best cost-structures and selection of CC service providers for a subset of the time period of the planning horizon. Decomposition of the demand is achieved through the boundary analysis of a continuous relaxation of the problem. Using the metrics defined in terms of the cost and time of completion, we demonstrate excellent performance in relation to optimal solutions. Our method reduced the computational time from hours to seconds for a representative 36-month problem and provided high-quality solutions (< 0.05% relative error). Given the importance of selecting resources and scheduling complex scientific workflows, we believe that this novel strategy will be beneficial for many researchers and users of cloud computing resources.

De La Torre, Luis↗
DOE OSTI2021

Off-the-shelf deep learning is not enough, and requires parsimony, Bayesianity, and causality

Abstract Deep neural networks (‘deep learning’) have emerged as a technology of choice to tackle problems in speech recognition, computer vision, finance, etc. However, adoption of deep learning in physical domains brings substantial challenges stemming from the correlative nature of deep learning methods compared to the causal, hypothesis driven nature of modern science. We argue that the broad adoption of Bayesian methods incorporating prior knowledge, development of solutions with incorporated physical constraints and parsimonious structural descriptors and generative models, and ultimately adoption of causal models, offers a path forward for fundamental and applied research.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Where did the tumor start? An inverse solver with sparse localization for tumor growth models

In this work, we present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction–diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of magnetic resonance imaging (MRI) scans drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. Like most cases in clinical practice, we use data from a single time snapshot. Moreover, the precise time relative to the initiation of the tumor is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the structure and magnitude of the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint on the initial condition of the tumor) and that the initial condition magnitude in the maximum norm is equal to one. We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over a previously existing solver that uses standard two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the two-norm regularized solver.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Geometric deep learning of RNA structure

RNA molecules adopt three-dimensional structures that are critical to their function and of interest in drug discovery. Few RNA structures are known, however, and predicting them computationally has proven challenging. We introduce a machine learning approach that enables identification of accurate structural models without assumptions about their defining characteristics, despite being trained with only 18 known RNA structures. The resulting scoring function, the Atomic Rotationally Equivariant Scorer (ARES), substantially outperforms previous methods and consistently produces the best results in community-wide blind RNA structure prediction challenges. By learning effectively even from a small amount of data, our approach overcomes a major limitation of standard deep neural networks. Because it uses only atomic coordinates as inputs and incorporates no RNA-specific information, this approach is applicable to diverse problems in structural biology, chemistry, materials science, and beyond.

Townshend, Raphael J. L.↗
DOE OSTI2023

Utilization of Additive Manufacturing for the Rapid Prototyping of C-Band Radiofrequency Loads

Additive manufacturing is a versatile technique that shows promise in providing quick and dynamic manufacturing for complex engineering problems. Research has been ongoing into the use of additive manufacturing for potential applications in radiofrequency (RF) component technologies. Here, we present a method for developing an effective prototype load produced from 316L stainless steel on a direct metal laser sintering machine. The model was tested using simulation software to verify the validity of the design. The load structure was manufactured by an online digital manufacturing company, showing the viability of using easily accessible tools to manufacture RF structures. The produced load was able to produce an S11 value of −22.8 dB at a C-band frequency of 5.712 GHz while under a vacuum. In a high-power test, the load was able to terminate a peak power of 8.1 MW. The discussion includes future applications of the present method and how it will help to improve the implementation of future accelerator concepts.

36 MATERIALS SCIENCE↗
DOE OSTI2024

A Novel Method to Train Classification Models for Structure Detection in In Situ Spacecraft Data

We present a method for creating spacecraft-like data which can be used to train Machine Learning (ML) models to detect and classify structures in in situ spacecraft data. First, we use the Grad-Shafranov equation to numerically solve for several magnetohydrostatic equilibria which are variations on a known analytic equilibrium. These equilibria are then used as the initial conditions for Particle-In-Cell simulations in which the structures of interest are observed and labeled. We then take one-dimensional slices through the simulations to replicate what a spacecraft collecting data from the simulation would observe. This sliced data then can be used as training data for the initial training of ML models intended for use on spacecraft data. We demonstrate the method applied to the problem of detecting small-scale plasmoids in the magnetotail, which is important for understanding complex magnetotail reconnection dynamics. The simple 1D classifier we train is able to detect more than 70% of the plasmoid points in the data set but also produces a large number of false positives. Our further work on this example problem is detailed, and further potential uses of the method are discussed.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

ForceFinder

SAND2025-11750O ForceFinder extends the Structural Dynamics Python Libraries (SDynPy) with comprehensive tools for inverse source estimation (ISE) tasks via frequency response function (FRF) matrix inversion. The software is designed for transfer path analysis and multiple-input/multiple-output (MIMO) vibration control problems. It allows users to estimate sources through various algorithms, from the basic Moore-Penrose pseudo-inverse to statistical learning methods such as Tikhonov regularization via an L-curve and elastic net regularization via an information criterion. ForceFinder uses an object-oriented framework, where all components of the ISE problem—such as FRFs, responses, and transformations—are stored in a "SourcePathReceiver" object. This software can be applied to any noise and vibration problem. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Carter, Steven [Sandia National Lab. (SNL-CA), Liv↗
DOE OSTI2020

Hyper-Differential Sensitivity Analysis of Uncertain Parameters in PDE-Constrained Optimization

Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional complexity, namely, uncertain parameters in the PDEs. Uncertainty quantification (UQ) is necessary to characterize, prioritize, and study the influence of these uncertain parameters. Sensitivity analysis, a classical tool in UQ, is frequently used to study the sensitivity of a model to uncertain parameters. In this article, we introduce "hyper-differential sensitivity analysis" which considers the sensitivity of the solution of a PDE-constrained optimization problem to uncertain parameters. Our approach is a goal-oriented analysis which may be viewed as a tool to complement other UQ methods in the service of decision making and robust design. We formally define hyper-differential sensitivity indices and highlight their relationship to the existing optimization and sensitivity analysis literatures. Assuming the presence of low rank structure in the parameter space, computational efficiency is achieved by leveraging a generalized singular value decomposition in conjunction with a randomized solver which converts the computational bottleneck of the algorithm into an embarrassingly parallel loop. Two multi-physics examples, consisting of nonlinear steady state control and transient linear inversion, demonstrate efficient identification of the uncertain parameters which have the greatest influence on the optimal solution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Multimodal, Multidimensional, and Multiscale X-ray Imaging at the National Synchrotron Light Source II

Over the last couple of decades, the synchrotron radiation research community has witnessed tremendous advancement in the field of x-ray imaging and microscopy. Continuing enhancement of the light sources’ brightness, advances in x-ray focusing optics, incorporation of precision instruments, and development of innovative imaging techniques are some of the leading contributors for the rapid progress. New imaging and microscopy beamlines, such as the ones at the National Synchrotron Light Source II (NSLS-II), a Department of Energy (DOE) Office of Science user facility located at DOE’s Brookhaven National Laboratory, are capable of performing more sophisticated and complicated measurements than ever before, either on their own or used together as a suite of tools. These sophisticated and complex measurements exhibit attributes for multimodal, multidimensional, and multiscale imaging. Recent popularity of these methods is strongly driven by the current trends in materials characterizations, where researchers desire to map out hierarchical materials structure over a large range of length scales, to understand structure-property correlation, to quantify materials structures in 3D, and/or to perform operando or in situ experiments. It is important to emphasize that “materials” under investigation are not only synthesized materials but also the natural materials. In this article, we describe multimodal, multidimensional and multiscale x-ray imaging capabilities of the NSLS-II beamlines and how these methods are used to tackle complex scientific problems.

47 OTHER INSTRUMENTATION↗
DOE OSTI2022

Diffusion Monte Carlo approaches for studying nuclear quantum effects in fluxional molecules

Abstract Diffusion quantum Monte Carlo (DMC) provides a powerful approach for obtaining the ground state energy and wave function of molecules, ions, and molecular clusters. The approach is uniquely well suited for studies of fluxional molecules, which undergo large amplitude vibrational motions even in their ground state. In contrast to the electronic structure problem, where the wave function must be antisymmetric with respect to exchange of any pair of electrons, the wave function for the ground vibrational state is nodeless. This greatly simplifies the application of DMC for vibrational problems. Because there is not a single potential function that can be used to describe the intramolecular and intermolecular interactions in all molecular systems, most methods that are used to describe nuclear quantum effects rely on a carefully chosen zero‐order description of the molecular vibrations. In contrast, DMC calculations can be performed in Cartesian coordinates, making the DMC algorithm easily transferable between different chemical systems. In this contribution, the theory that underlies DMC will be discussed along with important considerations for performing DMC calculations. Extensions for evaluating vibrationally excited states and molecular properties are also discussed. Insights that can be obtained from DMC calculations are illustrated in the context of the protonated water clusters. This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods Theoretical and Physical Chemistry > Spectroscopy

Chemistry↗
DOE OSTI2021

Reexamining the variational two-particle reduced density matrix for nuclear systems

We report that in most nuclear many-body methods, observables are calculated using many-body wave functions explicitly. The variational two-particle reduced density matrix method is one of the few exceptions to the rule. Ground-state energies of both closed-shell and open-shell nuclear systems can indeed be evaluated by minimizing a constrained linear functional of the two-particle reduced density matrix. However, it has virtually never been used in nuclear theory, because nuclear ground states were found to be well overbound, contrary to those of atoms and molecules. Consequently, we introduced new constraints in the nuclear variational two-particle reduced density matrix method, developed recently for atomic and molecular systems. Our calculations then show that this approach can provide a proper description of nuclear systems where only valence neutrons are included. For the nuclear systems where both neutrons and protons are active, however, the energies obtained with the variational two-particle reduced density matrix method are still overbound. The possible reasons for the noticed discrepancies and solutions to this problem will be discussed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗